For a left stochastic matrix on a network, principal eigenvectors associated with eigenvalue 1
have positive entries by Perron-Frobenius theorem, and they
have been used
to determine the influence of a vertex, see [1]}, [2]} and references therein.
Let \({\bf W}=(W(i,j))_{i,j\in V}\)  be the hyperlink matrix on a network described by a graph \({\mathcal {G}}=(V, E)\) , where weights
\(W(i,j)=0\)  for \((i,j)\notin E\)  and
\(W(i,j)=1/d_j\)  for \((i,j)\in E\) , the reciprocal of the degree \(d_j\)  of a vertex \(j\) . The matrix \({\bf W}\)  is a left stochastic matrix with
1 as the leading eigenvalue.
Applying the proposed PGDA to the hyperlink matrix \({\bf W}\) , we can locally evaluate principal eigenvectors of the hyperlink matrix and hence identify the local influence of a vertex on its neighborhood.
